9-Digit Numbers

Let R(X) be the number created by reversing the digits
of X. When a 9-digit Palindromic number P is divided by a 7-digit
number N, a remainder of R(N) is obtained. Also, dividing N by R(N)
gives a remainder D, where D is a prime number. The last digit of
neither P nor N is zero. Find any triplets (P,N,D) satisfying these
conditions. If in addition, the sum of the digits of D is a perfect
square, how many triplets exist?

Determine ten different nine digit decimal numbers
beginning with the same digit, such that their sum is divisible by nine of
the said numbers. How many possible solutions are there?

Determine four different nine digit decimal numbers
beginning with the same digit, such that their sum is divisible by three of
the said numbers. How many possible solutions are there?

Extension: Determine M+1 different N-digit decimal
numbers beginning with the same digit, such that their sum is divisible by M
of the said numbers, and N is as small as possible. For which M<10 do solutions exist?

Source: Adapted from K Sengupta of India. He does not yet have solutions for
all of these. Original Extension.